On multi-graded Proj schemes
Arnaud Mayeux, Simon Riche
TL;DR
The paper develops a comprehensive framework for Proj schemes graded by a finitely generated abelian group $M$, extending Grothendieck's Proj and recasting it through the Brenner--Schröer construction via a diagonalizable group action and the central notion of potions. It establishes the glueing mechanism (the magic of potions) for ${\rm Proj}^M(A)$, studies open subschemes defined by quasi-relevant elements, and extends Serre-type results to quasi-coherent sheaves in this multi-graded context, including twisting and derived-category equivalences. The framework is then applied to canonical geometric objects in Geometric Representation Theory: flag varieties, vector bundles over them, and the Springer resolution are realized as Proj schemes of natural multigraded rings, yielding canonical, choice-free constructions. The work also links Proj theory with dilatations and multi-centered blowups, providing a robust algebraic toolkit for relative Proj, base change, and geometric quotients, with broad implications for representation-theoretic geometry and beyond.
Abstract
We review the construction (due to Brenner--Schröer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for $\mathbb{N}$-graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the $\mathbb{N}$-graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner--Schröer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism.